Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

PARABOLIC-HYPERBOLIC SPDE

BENJAMIN GESS AND MARTINA HOFMANOV ´A

Abstract. We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a fullL¹ assuming no growth conditions on the nonlinearities. In addition, we prove a comparison result and anL¹-contraction property for the solutions.

1. Introduction

We study the regularity and well-posedness of quasilinear degenerate parabolic-hyperbolic SPDE of the form

du+ div(B(u))dt= div(A(u)∇u)dt+Φ(x, u)dW, x∈T^N, t∈(0, T), u(0) =u0,

(1.1)

whereW is a cylindrical Wiener process,u0∈L¹(T^N),B ∈C²(R,R^N), A∈C¹(R,R^N×N) takes values in the set of symmetric non-negative definite matrices and Φ(x, u) are Lipschitz continuous diffusion coefficients.

Equations of this form arise in a wide range of applications including the convection-diffusion of an ideal fluid in porous media. The addition of a stochastic noise is often used to account for numerical, empirical or physical uncertainties. In view these applications, we aim to treat (1.1) under general assumptions on the coefficientA, Band initial datau0. In particular, the coefficients are not necessarily linear nor of linear growth andAis not necessarily strictly elliptic. Hence, in particular, we include stochastic scalar conservation laws

du+ div(B(u))dt=Φ(x, u)dW and stochastic porous media equations

du+ div(B(u))dt= ∆u^[m]dt+Φ(x, u)dW, withm >2 andu^[m]:=sgn(u)u^m.

One of the main points of this paper is to provide a fullL¹ approach to (1.1). That is, we prove regularity estimates and well-posedness for (1.1) assuming no higher moments. More precisely, only u0 ∈ L¹(T^N) andno growth assumptions on the nonlinearitiesA, B are assumed. In particular, no Lipschitz continuity (and thus linear growth) assumptions onA, B are supposed. This causes severe difficulties: Firstly, the weak form of (1.1) is not necessarily well-defined since A(u), B(u) are not necessarily in L¹_loc(T^N) for u ∈ L¹(T^N). Therefore, renormalized solutions have to be considered (cf. [22, 9,1]). Secondly, in order to prove the uniqueness of L¹ entropy solutions an equi-integrability condition or, equivalently, a decay condition for the entropy defect measure is required (see a more detailed discussion below). The usual decay condition used in the deterministic case is not applicable in the stochastic case and a new condition and proof has to be found. Thirdly,

Date: November 4, 2016.

2010Mathematics Subject Classification. 60H15, 35R60.

Key words and phrases. quasilinear degenerate parabolic stochastic partial differential equation, kinetic formu- lation, kinetic solution, velocity averaging lemmas, renormalized solutions.

1

in the stochastic case, the usual proof of existence of entropy solutions relying on the Crandall- Liggett theory ofm-accretive operators inL¹(T^N) cannot be applied (cf. [13,12,8]). Instead, the construction of entropy solutions presented in this paper relies on new regularity estimates based on averaging techniques. The application of averaging techniques and the resulting regularity results are new for parabolic-hyperbolic SPDE of the type (1.1).

On the other hand,L¹(T^N) is a natural space to consider the well-posedness for SPDE of the type (1.1) since the operators div(B(·)), div(A(·)∇·) are accretive inL¹(T^N) (cf. the discussion of the e-property below). In addition, and in contrast to the deterministic case, restricting to bounded solutions and hence, by localization, to Lipschitz continuous coefficients A, B in (1.1) does not seem to be sensible in the stochastic case, since in general no uniformL^∞ bound will be satisfied by solutions to (1.1), due to the unboundedness of the driving noiseW.

As a particular example, (1.1) contains stochastic porous media equations (1.2) du= ∆u^[m]dt+Φ(x, u)dW, with m >2.

Stochastic porous media equations have attracted a lot of interest in recent years (cf. e.g. [49,48, 3,50] and the references therein). All of these results rely on anH⁻¹approach, that is, on treating

∆(·)^[m] as a monotone operator inH⁻¹. In contrast to the deterministic case, anL¹ approach to stochastic porous media equations had not yet been developed, since an analog of the concept of mild solutions in the Crandall-Liggett theory ofm-accretive operators (cf. [55, 8]) could not be found. However, theL¹framework offers several advantages: Firstly, more general classes of SPDE may be treated, secondly, contractive properties inL¹ norm are sometimes better than those in H⁻¹norm. We next address these points in more detail.

Concerning the class of SPDE, informally speaking, theH⁻¹ approach relies on applying (−∆)⁻¹ to (1.2) which then allows to use the monotonicity ofφ(u) :=u^[m] in order to prove the uniqueness of solutions. While this works well for the operator ∆φ(·), the reader may easily check that this approach fails in the presence of hyperbolic terms divB(u) as in (1.1) and can only be applied to reaction diffusion equations

(1.3) du= ∆u^[m]dt+f(u)dt+Φ(x, u)dW, withm >2.

under unnecessarily strong assumptions on the reaction termf (cf. e.g. [14,49] where (1.3) withf satisfying rather restrictive assumptions has been considered). Roughly speaking, the problem is that the Nemytskii operatoru7→f(u) is not necessarily monotone inH⁻¹ even iff is a monotone function. This changes drastically in the L¹ setting, since both u 7→ divB(u) and u 7→ f(u) are accretive operators onL¹ under relatively mild assumptions. In this paper, we resolve these issues by establishing a fullL¹approach to (1.1) based on entropy/kinetic methods. In particular, this extends available results on stochastic porous media equations by allowing hyperbolic terms divB(u) and our framework immediately¹extends to reaction termsu7→f(u) assuming only that f is weakly monotone and C².

We proceed by stating the main well-posedness result obtained in this paper, see Theorem 4.3, Theorem 4.9 below. The precise framework will be given in Section 2 below and for specific examples see Section2.4.

Theorem 1.1. Let u0 ∈L¹(T^N) and assume that A¹² is γ-H¨older continuous for some γ > ¹₂. Then, kinetic solutions to (1.1)are unique. Moreover, if u1, u2 are kinetic solutions to (1.1)with initial datau1,0 andu2,0, respectively, then

ess sup

t∈[0,T] Ek(u1(t)−u2(t))⁺k_L1(T^N)≤ k(u1,0−u2,0)⁺k_L1(T^N).

Assume in addition thatA, B satisfy a non-degeneracy assumption (cf. (2.3) below). Then there exists a unique kinetic solution u to (1.1) satisfying u ∈ C([0, T];L¹(T^N)), P-a.s., and for all

1We choose not to include the details on the treatment of reaction termsf(u) in this paper, since their treatment is similar to the noise terms Φ(u)dW.

p, q∈[1,∞) there exists a constantC >0 such that Eess sup

t∈[0,T]

ku(t)k^pq_Lp≤C(1 +ku0k^pq_Lp).

The second direction of advantages of the L¹ approach lies in dynamical properties. A natural question for stochastic porous media equations is their long-time behavior, that is, the existence and uniqueness of invariant measures, mixing properties etc. Ifu,vare two solutions to (1.2) with initial conditionsu0, v0 respectively, then

(1.4) Eku(t)−v(t)kH⁻¹≤e^Ctku0−v0kH⁻¹ ∀t≥0,

for some constant C >0. The constant C corresponds to the Lipschitz norm of u7→ Φ(u) as a map fromH⁻¹ toL₂(U;H⁻¹). In particular, the dynamics induced by (1.2), in general, will not be non-expanding inH⁻¹. In contrast, we show that

Eku(t)−v(t)kL¹ ≤ ku0−v₀kL¹ ∀t≥0,

that is, in the L¹ setting we can choose the constant C in (1.4) to be zero. In particular, this implies the e-property (cf. [40]) for the associated Markovian semigroup P_tf(x) := Ef(X_t^x) on L¹(T^N). The e-property has proven vital in the proof of existence and uniqueness of invariant measures for SPDE with degenerate noise (cf. [31,30,24,40]).

Forx∈L¹(T^N) let

Px:= (u^x_·)_∗P,

that is,Pxis the law ofu^x_· onC([0,∞);L¹(T^N)), whereu^x_· denotes the kinetic solution to (1.1) with initial condition x. We equip C([0,∞);L¹(T^N)) with the canonical filtration Gt and evaluation mapsπt(w) :=w(t) forw∈C([0,∞);L¹(T^N)),t≥0. As in [15], using Theorem1.1, we obtain Corollary 1.2. The family{Px}_x∈L¹ is a time-homogeneous Markov process onC([0,∞);L¹(T^N)) with respect toGt, i.e.

Ex(F(π_t+s)|G_s) =Eπs(F(π_t)) P_x-a.s.

In addition,{Px}_x∈L1(T^N) is Feller and satisfies thee-property (cf. [40]).

As mentioned above, we prove new regularity estimates for kinetic solutions to (1.1) of the type u(t)∈W^α,1(T^N) for a.e. (ω, t),

for someα >0, based on stochastic velocity averaging lemmas. Even in the case of pure stochastic porous medium equations (1.2) this extends previously available regularity results. For related deterministic results, see [7, 23,38, 52], for stochastic hyperbolic conservation laws see [17]. Our approach is mainly based on [52], but substantial difficulties due to the stochastic integral have to be overcome. Indeed, in most of the deterministic results (with the notable exception of [7]), the time variable does not play a special role and is regarded as another space variable and, in particular, space-time Fourier transforms are employed in the proofs. This changes in the stochastic case due to the irregularity of the noise in time. Therefore, it was argued in [17] that these methods are not suitable for the stochastic case and instead the approach of [7] which does not rely on Fourier transforms in time was employed. In the present paper, we rely on different arguments:

We put forward averaging lemmas that rely on space-time Fourier transforms, Littlewood-Paley decomposition and a careful analysis of each of the appearing terms. As a consequence, we are able to estimate the stochastic integral as well as the kinetic measure term directly by averaging techniques, without any additional damping (as compared to [7, 17]). Moreover, our averaging lemmas apply to the case of nonhomogeneous equations, that is, PDEs with zero, first and second order terms andmultiplicative noise.

More precisely, as a corollary of our main regularity result for (1.1), see Theorem3.1and Corollary 3.3below, we obtain

Theorem 1.3. Assume that A, B satisfy a non-degeneracy condition (cf. (2.3)below) and are of polynomial growth of orderp. Letube a kinetic solution to (1.1). Then

Ekuk_L1([0,T];W^s,1(T^N)) .ku0k^2p+3

L^2p+3_x + 1, for somes >0.

We now proceed with the announced more detailed discussion of the comparison to the proof of well-posedness of entropy solutions for deterministic parabolic-hyperbolic PDE (cf. [11])

(1.5) du+ div(B(u))dt= div(A(u)∇u)dt.

The inclusion of stochastic perturbation causes several additional difficulties. First, the proof of existence of solutions in [11] relies on the (simple) proof ofBV regularity of solutions to (1.5). Such aBV estimate is not known in the stochastic case and does not seem to be easy to obtain (for a discussion of the necessity of such estimates in the construction of a solution see Section1.5below).

Therefore, we instead rely on regularity obtained based on averaging techniques. Second, the equi- integrability estimates encoded in the decay properties of the kinetic measure in the deterministic situation, that is, in the assumption (cf. [11, Definition 2.2 (iv)])

|ξ|→∞lim Z

m(t, x, ξ)dtdx= 0

do not seem to be suitable in the stochastic case, since themultiplicativenoise term Φ(x, u) is less well behaved in terms of these estimates. Indeed, the corresponding proof of a-priori estimates proceeds along different lines than in the deterministic case (cf. Proposition4.7below). Therefore, we replace these decay estimates by the weaker decay condition

(1.6) lim

`→∞

1 2^`E

Z

1_|ξ|≥2`m(t, x, ξ)dtdxdξ= 0

and prove the uniqueness of kinetic solutions under this weaker assumption.

The kinetic approach to (deterministic) scalar conservation laws was introduced by Lions, Perthame, Tadmor in [45] and extended to parabolic-hyperbolic PDE in [11], including PDE of porous media type. In the stochastic case, the well-posedness of such PDE had not previously been shown. Under more restrictive assumptions, namely high moment bounds u₀ ∈ T

p≥1L^p(Ω;L^p(T^N)), bounded- ness of the diffusion matrixAand polynomial growth ofB⁰⁰, the well-posedness was shown in [16].

In particular, neither porous media equations nor general L¹ initial data could be handled. In contrast, besides providing a fullL¹well-posedness theory, we only assume Ato be locally H¨older continuous (cf. (2.2) below) and completely remove the growth assumptions onA, B.

Special cases of SPDE of the type (1.1) have attracted a lot of interest in recent years. For de- terministic hyperbolic conservation laws, see [6, 37, 41, 44, 45, 46, 47]. Stochastic degenerate parabolic equations were studied in [4,16, 34] and stochastic conservation laws in [5, 10, 17,18, 19,25,33,36,39,51,54]. Recently, also scalar conservation laws driven by rough paths have been considered in [26,35,20]. Other types of stochastic scalar conservation laws, for which randomness enters in form of a random flux have been considered in [42, 43, 29, 27]. Stochastic quasilinear parabolic-hyperbolic SPDE with random flux have been considered in [28].

The paper is organized as follows. In Section2, we introduce the precise framework and the concept of kinetic solutions. Our main regularity result will be proven in Section3. This is then used in Section4to prove the well-posedness for kinetic solutions.

2. Preliminaries

2.1. Notation. In this paper, we use the bracketsh·,·ito denote the duality between the space of distributions overT^N×RandC_c^∞(T^N×R) and the duality betweenL^p(T^N×R) andL^q(T^N×R).

If there is no danger of confusion, the same brackets will also denote the duality betweenL^p(T^N) and L^q(T^N). By M([0, T]×T^N ×R) we denote the set of Radon measures on [0, T]×T^N ×R

andM⁺([0, T]×T^N ×R) then contains nonnegative Radon measures andMb([0, T]×T^N ×R) contains finite measures. We also use the notation

n(φ) = Z

[0,T]×T^N×R

φ(t, x, ξ) dn(t, x, ξ), n∈ M([0, T]×T^N×R), φ∈Cc([0, T]×T^N ×R).

In order to signify thatn ∈ M([0, T]×T^N ×R) is only considered on [0, T]×T^N ×D for some compact setD⊂Rwe writen1_D. In particular,

kn1DkMt,x,ξ= Z

[0,T]×T^N×D

d|n|(t, x, ξ).

The differential operators of gradient∇, divergence div and Laplacian ∆ are always understood with respect to the space variablex. For two matricesA, B of the same size we set

A:B :=X

ij

aijbij.

Throughout the paper, we use the termrepresentativefor an element of a class of equivalence.

Finally, we use the letterCto denote a generic constant that might change from one line to another.

We also employ the notationx.yif there exists a constantCindependent of the variables under consideration such thatx≤Cyand we writex∼y ifx.y andy.x.

2.2. Setting. We now give the precise assumptions on each of the terms appearing in the above equation (1.1). We work on a finite-time interval [0, T], T > 0, and consider periodic boundary conditions:x∈T^N whereT^N =R^N|(2πZ^N) is theN-dimensional torus. For the fluxBwe assume (2.1) B= (B1, . . . , BN)∈C²(R,R^N)

and we set b = ∇B. The diffusion matrix A = (A_ij)^N_i,j=1 ∈ C¹(R;R^N×N) is assumed to be symmetric, positive semidefinite and its square root σ :=A¹² is assumed to be locally γ-H¨older continuous for someγ >1/2, that is, for allR >0 there is a constant C=C(R) such that (2.2) |σ(ξ)−σ(ζ)| ≤C(R)|ξ−ζ|^γ ∀ξ, ζ∈R,|ξ|,|ζ| ≤R.

We will further require a non-degeneracy condition for the symbolLassociated to the kinetic form of (1.1)

L(iu, in, ξ) :=i(u+b(ξ)·n) +n^∗A(ξ)n.

ForJ, δ >0 andη∈C_b^∞(R) nonnegative let ω_L^η(J;δ) := sup

u∈R,n∈Z^N

|n|∼J

|Ω^η_L(u, n;δ)|, Ω^η_L(u, n;δ) :={ξ∈suppη; |L(iu, in, ξ)| ≤δ}

and Lξ := ∂ξL. We suppose that there exist α ∈ (0,1), β > 0 and a measurable map ϑ ∈ L^∞_loc(R; [1,∞)) such that

ω_L^η(J;δ).η

δ J^β

^α

sup

u∈R,n∈Z^N

|n|∼J

sup

ξ∈suppη

|Lξ(iu, in, ξ)|

ϑ(ξ) .^ηJ^β, ∀δ >0, J&1.

(2.3)

The requirement of a suitable non-degeneracy condition is classical in the theory of averaging lemmas and therefore it will be essential for Theorem 3.1. The localization η and the weight ϑ give two possibilities to control the growth of L_ξ in ξ. In the proof of existence in Subsection 4.2.3, we employ (2.3) withϑ≡1 and η compactly supported which allows to obtain regularity of the localized averageR

Rχu(ξ)η(ξ)dξ without any further integrability assumptions on u. On the contrary, with a suitable choice of ϑ, we may considerη ≡1 to obtain regularity of uitself provided it possesses certain additional integrability. We refer the reader to Subsection 2.4 for further discussion of (2.3) as well as for application to particular examples.

Regarding the stochastic term, let (Ω,F,(Ft)t≥0,P) be a stochastic basis with a complete, right- continuous filtration. LetP denote the predictable σ-algebra on Ω×[0, T] associated to (Ft)t≥0. The initial datum u0 is F0-measurable and the process W is a cylindrical Wiener process, that is, W(t) =P

k≥1β_k(t)e_k with (β_k)_k≥1 being mutually independent real-valued standard Wiener processes relative to (Ft)t≥0 and (ek)k≥1 a complete orthonormal system in a separable Hilbert spaceU. In this setting we can assume without loss of generality that theσ-algebraF is countably generated and (Ft)_t≥0 is the filtration generated by the Wiener process and the initial condition.

For eachz∈L²(T^N) we consider a mapping Φ(z) :U→L²(T^N) defined byΦ(z)ek =gk(·, z(·)).

We suppose thatgk ∈C(T^N×R) and there exists a sequence (αk)k≥1of positive numbers satisfying D:=P

k≥1α²_k<∞such that

|gk(x,0)|+|∇xgk(x, ξ)|+|∂ξgk(x, ξ)| ≤αk, ∀x∈T^N, ξ∈R. (2.4)

Note that it follows from (2.4) that

(2.5) |gk(x, ξ)| ≤α_k(1 +|ξ|), ∀x∈T^N, ξ∈R. and

(2.6) X

k≥1

|gk(x, ξ)−g_k(y, ζ)|²≤C |x−y|²+|ξ−ζ|²

, ∀x, y∈T^N, ξ, ζ∈R.

Consequently, denotingG²(x, ξ) =P

k≥1|gk(x, ξ)|²it holds

G²(x, ξ)≤2D(1 +|ξ|²) ∀x∈T^N, ξ∈R. The conditions imposed onΦ, particularly assumption (2.4), imply that

Φ:L²(T^N)−→L2(U;L²(T^N)),

whereL2(U;L²(T^N)) denotes the collection of Hilbert-Schmidt operators fromUtoL²(T^N). Thus, given a predictable processu∈L²(Ω;L²(0, T;L²(T^N))), the stochastic integralt7→Rt

0Φ(u)dW is a well defined process taking values inL²(T^N) (see [15] for a detailed construction).

Finally, we define the auxiliary spaceU0⊃Uvia U0=

v=X

k≥1

αkek; X

k≥1

α²_k k² <∞

, endowed with the norm

kvk²_U0 =X

k≥1

α²_k

k², v=X

k≥1

αkek.

Note that the embeddingU,→U0 is Hilbert-Schmidt. Moreover, trajectories of W areP-a.s. in C([0, T];U0) (see [15]).

2.3. Kinetic solutions. Let us introduce the definition of kinetic solution as well as the related definitions used throughout this paper. It is a generalization of the concept of kinetic solution studied in [16], which is suited for establishing well-posedness in the L¹-framework, that is, for initial conditions inL¹(Ω;L¹(T^N)). In that case, the corresponding kinetic measure is not finite and one can only prove suitable decay at infinity.

Definition 2.1 (Kinetic measure). A mapping m from Ω to M⁺([0, T]×T^N ×R), the set of nonnegative Radon measures over [0, T]×T^N×R, is said to be a kinetic measure provided

(i) For allψ∈Cc([0, T)×T^N×R), the process Z

[0,t]×T^N×R

ψ(s, x, ξ) dm(s, x, ξ) is predictable.

(ii) Decay ofmfor largeξ: it holds true that

`→∞lim 1

2<sup>`</sup>Em(A<sub>2</sub>`) = 0, where

A₂` = [0, T]×T<sup>N</sup> × {ξ∈R; 2<sup>`</sup>≤ |ξ| ≤2^`+1},

Definition 2.2(Kinetic solution). A mapu∈L¹(Ω×[0, T],P,dP⊗dt;L¹(T^N)) is called a kinetic solution to (1.1) with initial datum u0 if the following conditions are satisfied

(i) For allφ∈C_c^∞(R),φ≥0, div

Z u 0

φ(ζ)σ(ζ) dζ∈L²(Ω×[0, T]×T^N).

(ii) For allφ1, φ2∈C_c^∞(R),φ1, φ2≥0, the following chain rule formula holds true (2.7) div

Z u 0

φ₁(ζ)φ₂(ζ)σ(ζ) dζ=φ₁(u) div Z u

0

φ₂(ζ)σ(ζ) dζ in L²(Ω×[0, T]×T^N).

(iii) Letn1: Ω→ M⁺([0, T]×T^N×R) be defined as follows: for allϕ∈C_c^∞([0, T]×T^N×R), ϕ≥0,

(2.8) n₁(ϕ) =

Z T 0

Z

T^N

div Z u

0

pϕ(t, x, ζ)σ(ζ) dζ

2

dxdt.

There exists a kinetic measurem≥n₁,P-a.s., such that the pair (f =1_u>ξ, m) satisfies, for allϕ∈C_c^∞([0, T)×T^N ×R),P-a.s.,

Z T 0

f(t), ∂tϕ(t) dt+

f0, ϕ(0) +

Z T 0

f(t), b· ∇ϕ(t) dt+

Z T 0

f(t), A: D²ϕ(t) dt

=−X

k≥1

Z T 0

Z

T^N

gk x, u(t, x)

ϕ t, x, u(t, x)

dxdβk(t)

−1 2

Z T 0

Z

T^N

G² x, u(t, x)

∂ξϕ t, x, u(t, x)

dxdt+m(∂ξϕ).

(2.9)

The definition of a kinetic solution given in Definition 2.2 generalizes the definition of kinetic solutions given in [16, Definition 2.2] which applies to the case of high integrability, that is, for u∈L^p(Ω;L^p([0, T]×T^N)) for all p≥1.

Remark 2.3. Letu∈L^p(Ω;L^p([0, T]×T^N)) for allp≥1. Then,uis a kinetic solution to (1.1) in the sense of [16, Definition 2.2] if and only ifuis a kinetic solution in the sense of Definition2.2.

Remark 2.4. We emphasize that a kinetic solution is, in fact, a class of equivalence in L¹(Ω× [0, T];L¹(T^N)) so not necessarily a stochastic process in the usual sense. The term representative is then used to denote an element of this class of equivalence.

Let us conclude this section with two related definitions.

Definition 2.5 (Young measure). Let (X, λ) be a finite measure space. A mapping ν fromX to the set of probability measures onRis said to be a Young measure if, for allψ∈Cb(R), the map z7→νz(ψ) from X intoR is measurable. We say that a Young measureν vanishes at infinity if, for allp≥1,

Z

X

Z

R

|ξ|^pdν_z(ξ) dλ(z)<∞.

Definition 2.6(Kinetic function). Let (X, λ) be a finite measure space. A measurable function f :X×R→[0,1] is said to be a kinetic function if there exists a Young measureν onX vanishing at infinity such that, forλ-a.e. z∈X, for allξ∈R,

f(z, ξ) =νz(ξ,∞).

Remark 2.7. Note, that iff is a kinetic function then∂ξf =−ν forλ-a.e. z∈X. Similarly, letu be a kinetic solution of (1.1) and consider f =1u>ξ. We have ∂ξf =−δu=ξ, where ν=δu=ξ is a Young measure on Ω×[0, T]×T^N. Throughout the paper, we will often write νt,x(ξ) instead of δ_u(t,x)=ξ.

2.4. Applications. In this section we consider the model example of a convection-diffusion SPDE with polynomial nonlinearities, that is, letN = 1 and consider

du+∂x

u^k k

dt=∂x |u|^m−1∂xu

dt+Φ(x, u)dW, i.e. (1.1) withb(ξ) =B⁰(ξ) =ξ^k−1, A(ξ) =|ξ|^m−1, fork≥2,m >2.Hence,

L(iu, in, ξ) =i(u+ξ^k−1n) +|ξ|^m−1n², and

(2.10) |Lξ(iu, in, ξ)|.|ξ|^k−2|n|+|ξ|^m−2n². Forη∈C_b^∞(R) andu∈R, n∈Z, |n| ∼J we consider

Ω^η_L(u, n;δ) ={ξ∈suppη;|i(u+ξ^k−1n) +|ξ|^m−1n²| ≤δ}

and observe

Ω^η_L(u, n;δ)⊂Ω_A∩Ω_b, where

ΩA:={ξ∈suppη;|ξ|^m−1|n|²≤δ}, Ω_b:={ξ∈suppη;|i(u+ξ^k−1n)| ≤δ}.

Note that the set Ω_A is localized around 0 in the sense that ΩA=

(

ξ∈suppη;|ξ| ≤ δ

J²

m−1¹ ) , whereas the set Ω_b is moving according to the value ofu:

Ωb= (

ξ∈suppη;

u−δ J

_k−1¹

≤ξ≤

u+δ J

_k−1¹ ) .

In view of the second part of the condition (2.3) we chooseβ = 2 whenever a second order operator is present. Therefore we setβ= 2 andα= _m−1¹ , which yields the first part of (2.3) independently ofη

(2.11) ω_L^η(J;δ).

δ J²

_m−1¹ .

Regarding the second condition, it is necessary to control theξ-growth in (2.10). Our formulation of the nondegeneracy condition (2.3) offers two ways of doing so: either using a (compactly supported) localizationηor a weightϑ. Using the first approach, Theorem3.1yields regularity of the localized average ¯η(u) =R

Rχ_u(t,x)(ξ)η(ξ)dξ without any further integrability assumptions on the solution u. On the other hand, the second approach allows to obtain regularity of the solutionuitself, i.e.

settingη ≡1, but requires higher integrability ofu. To be more precise, in the case of (2.10) we setϑ(ξ) = 1 +|ξ|^k∨m−2 and assume thatu∈L^p(Ω×[0, T]×T^N) forp= 2(k∨m−2) + 3.

In the case of a purely hyperbolic equation with a polynomial nonlinearityb(ξ) =ξ^k−1,k≥2,we obtain

Ω^η_L(u, n;δ) =

ξ∈suppη; u−δ

J ≤ξ^k−1≤u+δ J

,

which implies the first condition in (2.3) independently ofη withα= _k−1¹ ,β= 1. For the second condition we proceed the same way as above.

3. Regularity

In this section we establish a regularity result for solutions to (1.1), based on averaging techniques.

Throughout this section we use the following notation: for a kinetic solution u, let χ := χu = 1u>ξ−10>ξ.Then we have, in the sense of distributions,

(3.1) ∂tχ+b(ξ)· ∇χ−A(ξ) :D²χ=∂ξq−

∞

X

k=1

(∂ξχ)gkβ˙k+

∞

X

k=1

δ0gkβ˙k,

whereq=m−¹₂G²δ_u=ξ.Forη ∈C_b^∞(R) let ¯η∈C^∞ be such that ¯η⁰=η and ¯η(0) = 0. We then have

¯ η(u) =

Z

R

χ_u(t,x)(ξ)η(ξ) dξ.

Theorem 3.1. Assume (2.1), (2.4). Let η ∈ C_b^∞(R;R+) and assume that there are α ∈ (0,1), β >0 and a measurable map ϑ ∈ L^∞_loc(R; [1,∞)) such that (2.3) is satisfied. Let Θη : R → R⁺ such thatΘ_η⁰ = (|ξ|²+ 1)ϑ²(ξ)(η(ξ) +|η⁰|(ξ)). Ifuis a kinetic solution to (1.1)then

¯ η(u) =

Z

R

χ_u(t,x)(ξ)η(ξ) dξ∈L^r(Ω×[0, T];W^s,r(T^N)), s < α²β 6(1 + 2α). with ¹_r >^1−θ₂ +^θ₁,θ=_4+α^α and

(3.2)

kη(u)k¯ _Lr(Ω×[0,T];W^s,r(T^N)).^ηkη(|u¯ 0|)k^1/2_L1

ω,x+kΘη(|u|)k^1/2_L1

ω,t,x+ sup

0≤t≤T

k¯η(|u|)kL¹_ω,x

+kmϑ(η+|η⁰|)kL¹_ωMt,x,ξ+ 1.

where the constant in the inequality depends onb andA via the constants appearing in (2.3)only and onη only via itsC¹ norm.

Remark 3.2. Ifη is compactly supported, then we may always takeϑ≡1 in (2.3). Furthermore, in this case the right hand side in (3.2) is always finite.

In order to deduce regularity foruitself we chooseη≡1. Ifϑis a polynomial of orderp, thenΘ_η is a polynomial of order 2p+ 3 and by Lemma4.6below we havekΘη(|u|)k_L¹²1

ω,t,x.ku0k

2p+3 2

L^2p+3_ω,t,x+ 1 andkmϑkL¹_ωMt,x,ξ .ku0k^p+2_Lp+2+ 1. In conclusion, we obtain

Corollary 3.3. Suppose (2.3)is satisfied forη≡1 andϑbeing a polynomial of orderp. Letube the kinetic solution²to (1.1). Then

kuk_Lr(Ω×[0,T];W^s,r(T^N)) .ku₀k^2p+3

L^2p+3ω,x

+ 1.

Proof of Theorem 3.1. The proof proceeds in several steps. In the first step, the solutionχ=χ_u= f−1_0>ξ is decomposed into Littlewood-Paley blocksχ_J and subsequently each Littlewood-Paley block is decomposed according to the degeneracy of the symbolL(iu, in, ξ). This decomposition off serves as the basis of the following averaging techniques. In the second step, each part of the decomposition is estimated separately, relying on the non-degeneracy condition (2.3). In the last step, these estimates are combined and interpolated in order to deduce the regularity off. The principle idea of the above decomposition off follows [52]. However, the stochastic integral in (1.1) leads to additional difficulties and requires a different treatment of the time-variable. This is resolved here by passing to the mild form (cf. (3.4) below) and then estimating all occurring terms separately, interpolating the estimates in the end.

2Well-posedness of kinetic solutions to (1.1) is proved in Section4below.

Decomposition ofχ. We introduce a cut-off in time, that is, letφ=φ^λ∈C¹([0,∞)) such that 0≤φ≤1,φ≡1 on [0, T −λ], φ≡0 on [T,∞) and|∂tφ| ≤ _λ¹ for someλ∈(0,1) to be eventually sent to 0. For notational simplicity, we omit the superscriptλin the following computations and let it only reappear at the end of the proof, where the passage to the limit inλis discussed.

Then,χφsolves, in the sense of distributions,

(3.3) ∂_t(χφ) +b(ξ)· ∇(χφ)−A(ξ) :D²(χφ) =∂_ξ(φq)−

∞

X

k=1

∂_ξ(χφ)g_kβ˙_k+

∞

X

k=1

δ₀φg_kβ˙_k+χ∂_tφ.

Next, we decomposeχinto Littlewood-Paley blocks χJ, such that the Fourier transform in space χcJ is supported by frequencies|n| ∼J forJ dyadic. This is achieved by taking a smooth partition of unity 1≡ ϕ0(z) +P

J&1ϕ(J⁻¹z) such that ϕ0 is a bump function supported inside the ball

|z| ≤2 andϕis a bump function supported in the annulus ¹₂≤ |z| ≤2, and setting χ0(t, x, ξ) :=F_x⁻¹

ϕ0(n) ˆχ(t, n, ξ) (x), χ_J(t, x, ξ) :=F_x⁻¹

ϕn

J

χ(t, n, ξ)ˆ

(x), J &1.

This leads to the decomposition

χ=χ0+X

J&1

χJ.

The regularity ofχ0being trivial, we only focus on the estimate of χJ forJ &1. Localizing (3.3) in Littlewood-Paley blocks yields

∂t(χJφ) +b(ξ)· ∇(χJφ)−A(ξ) :D²(χJφ) =∂ξ(φqJ)−

∞

X

k=1

∂ξ(χφgk)Jβ˙k+

∞

X

k=1

(χφ∂ξgk)Jβ˙k

+

∞

X

k=1

δ0φ(gk)Jβ˙k+χJ∂tφ.

After a preliminary step of regularization, we may test by S^∗(T −t)ϕ for ϕ ∈ C(T^N) in (3.1), whereS(t) denotes the solution semigroup to the linear operator

χ7→b(ξ)· ∇χ−A(ξ) :D²χ.

This leads to the mild form (χ_Jφ)(t) =S(t)χ_J(0) +

Z t 0

S(t−s)∂_ξ(φq_J) ds−

∞

X

k=1

Z t 0

S(t−s)∂_ξ(g_kχφ)_Jdβ_k(s)

+

∞

X

k=1

Z t 0

S(t−s)((∂ξgk)χφ)Jdβk(s) +

∞

X

k=1

Z t 0

S(t−s)δ0φgk,Jdβk(s) (3.4)

+ Z t

0

S(t−s)χJ∂tφds, where we have used

ϕn J

Fx

Z t 0

S(t−s)∂ξ(χφ)gkdβk(s)

(n)

=ϕn J

Z t 0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)∂_ξF_x(g_kχφ)(s, n, ξ) dβ_k(s)

−ϕn J

Z t 0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)Fx((∂ξgk)χφ)(s, n, ξ) dβk(s)

= Z t

0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)∂ξFx(gkχφ)J(s, n, ξ) dβk(s)

− Z t

0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)F_x((∂_ξg_k)χφ)_J(s, n, ξ) dβ_k(s)

and

ϕn J

Fx

Z t 0

S(t−s)δ0φgkdβk(s)

(n)

=ϕn J

Z t 0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)δ0φgbk(n, ξ) dβk(s)

= Z t

0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)δ0φgdk,J(n, ξ) dβk(s).

ForJ & 1 fixed, we next decompose the action inξ-variable according to the degeneracy of the operatorL(iu, in, ξ).Namely, forKdyadic, let 1≡ψ0(z) +P

K&1ψ1(K⁻¹z) be a smooth partition

of unity such thatψ0is a bump function supported inside the ball|z| ≤2 andψ1is a bump function supported in the annulus ¹₂ ≤ |z| ≤2, and write

1_0≤t(χJφ)(t, x, ξ) =F_tx⁻¹

ψ0

L(iu, in, ξ) δ

Ftx

1_0≤t(χJφ) (u, n, ξ)

(t, x)

+X

K&1

F_tx⁻¹

ψ1

L(iu, in, ξ) δK

Ftx

10≤t(χJφ) (u, n, ξ)

(t, x)

=:χ⁽⁰⁾_J (t, x, ξ) +X

K&1

χ^(K)_J (t, x, ξ).

Hence, we consider the decomposition

1_0≤tχφ=1_0≤t

χ₀+X

J&1

χ_J

φ=1_0≤tχ₀φ+X

J&1



χ⁽⁰⁾_J (t, x, ξ) +X

K&1

χ^(K)_J (t, x, ξ)



.

Since ψ₀ is supported at the degeneracy, we will apply a trivial estimate. However, ψ₁ is sup- ported away from the degeneracy and therefore we may use the equation and the non-degeneracy assumption (2.3). From (3.4) we obtain

χ^(K)_J (t, x, ξ) =F_tx⁻¹ψ1

L(iu, in, ξ) δK

Ftx

1_0≤tS(t)χ0,J +1_0≤t Z t

0

S(t−s)∂ξ(φqJ) ds

−1_0≤t

∞

X

k=1

Z t 0

S(t−s)∂ξ(gkχφ)Jdβk(s) +1_0≤t

∞

X

k=1

Z t 0

S(t−s)((∂ξgk)χφ)Jdβk(s)

+1_0≤t

∞

X

k=1

Z t 0

S(t−s)δ0φgk,Jdβk(s) +1_0≤t Z t

0

S(t−s)χJ∂tφds

(t, x).

Multiplying the above byη∈C_b^∞(R) and integrating overξ∈R, we set Z

R

χ^(K)_J (t, x, ξ)η(ξ)dξ=:I₁+I₂−I₃+I₄+I₅+I₆

and we estimate the right hand side term by term below. Note that since Ftx

1_0≤t

Z t 0

S(t−s)∂_ξ(g_kχφ)_Jdβ_k(s)

=Ft

10≤t

Z t 0

e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)∂ξ(g\kχφ)_J(s, ξ, n) dβk(s)

= 1

(2π)^1/2 Z Z

1_0≤s≤te−(ib(ξ)·n+n^∗A(ξ)n)(t−s)∂_ξ(g\_kχφ)_Jdβ_k(s) e^−itudt

= 1

(2π)^1/2 Z Z

1_t−s≥0e−(ib(ξ)·n+n^∗A(ξ)n)(t−s)e^−itudt1_0≤s∂ξ(g\kχφ)_J(s, ξ, n) dβk(s)

= Z

1_r≥0e−(ib(ξ)·n+n^∗A(ξ)n)re^−irudr 1 (2π)^1/2

Z ∞ 0

∂_ξ(g\_kχφ)_J(s, ξ, n)e^−isudβ_k(s)

= 1

i(u+b(ξ)·n) +n^∗A(ξ)n 1 (2π)^1/2

Z ∞ 0

∂ξ(g\kχφ)_J(s, ξ, n)e^−isudβk(s), we have

I3= 1 (2π)^1/2

1 (δK)

Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

^∞ X

k=1

Z ∞ 0

e^−isu∂ξ(g\kχφ)_J(s, ξ, n) dβk(s)

η(ξ)dξ,

where

ψ(z) :=˜ ψ₁(z)/z.

We argue similarly for the remaining termsI_i, e.g. forI₂we note that F_tx

1_0≤t

Z t 0

S(t−s)∂_ξ(φq_J) ds

= 1

i(u+b(ξ)·n) +n^∗A(ξ)nF_tx[1_0≤t∂_ξ(φq_J)]

EstimatingIi,i= 1, . . . ,6.

Estimate ofI1. Using Plancherel and H¨older’s inequality we observe kI1k²_L2

t,x= 1

2π(δK)² Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

ˆ

χ_J(0, n, ξ)

η(ξ)dξ

2

L²_t,x

= 1

2π(δK)² Z

u

X

n

Z

R

ψ˜

L(iu, in, ξ) δK

ˆ

χJ(0, n, ξ)η(ξ)dξ

2

du

≤ 1

2π(δK)² Z

u

X

n

Z

R

ψ˜

L(iu, in, ξ) δK

2

1suppηdξ

× Z

R

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}

χˆJ(0, n, ξ)

2η²(ξ)dξdu.

Then using (2.3) and Z

u

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}du≤ Z

u

1_{|u|²<(2δK)²}du.δK (3.5)

we obtain

kI1k²_L2 t,x. 1

δK δK

J^β α

kχJ(0)ηk²_L2 x,ξ. (3.6)

Estimate ofI2. First, we integrate by parts to obtain kI2k_L1

tW_x^−,q = 1 δK

Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

Ftx(1_0≤tφ∂ξqJ)

η(ξ)dξ _L1

tW_x^−,q

≤ 1 (δK)²

Z

R

F_tx⁻¹

ψ˜⁰

L(iu, in, ξ) δK

Lξ(iu, in, ξ)

ϑ(ξ) Ftx(10≤tφϑqJ)(u, ξ, n)

η(ξ)dξ _L1

tWx^−,q

+ 1

(δK)² Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

F_tx(1_0≤tφq_J)(u, ξ, n)

η⁰(ξ)dξ _L1

tW_x^−,q

We apply CorollaryA.4to estimate the second term on the right hand side. For the first one, we first note that (2.3) implies that (for simplicity restricting to the caseβ = 2 while β = 1 can be handled analogously)

|b⁰_i(ξ)|

ϑ(ξ) .^η|n|, |A⁰_ij(ξ)|

ϑ(ξ) .^η 1, i, j∈ {1, . . . , N}.

SinceLξ(iu, in, ξ) is a polynomial inn, we may apply [2, Lemma 2.2] to deduce that m(n, ξ) :=Lξ(iu, in, ξ)

ϑ(ξ) = ib⁰(ξ)·n+n^∗A⁰(ξ)n ϑ(ξ)

localized to |n| ∼ J, ξ ∈ suppη, is an L¹-Fourier multiplier with norm bounded by J^β. Revis- iting the proofs of Lemma A.3 and Corollary A.4 with the multiplier ψ_m(u,n,ξ)

δ

replaced by ψ_m(u,n,ξ)

δ

m(n, ξ) then allows to estimate the first term on the right hand side to obtain

kI2k_L1

tWx^−,q . 1

(δK)²J^βkφϑqJηkM_t,x,ξ+ 1

(δK)²kφqJη⁰kM_t,x,ξ . 1

(δK)²J^βkφϑqJ(η+|η⁰|)kM_t,x,ξ, where

N

q⁰ < <1< q< N N− andis chosen sufficiently small. Consequently,

kI2k_L1

t,x. 1

(δK)²J^β+kφϑqJ(η+|η⁰|)k_Mt,x,ξ. (3.7)

Estimate ofI3. Using Plancherel and Itˆo’s formula, we note that EkI3k²_L2

t,x= 1

2π(δK)²E Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

^∞ X

k=1

Z ∞ 0

e^−isu∂_ξ(g\_kχφ)_J(s, ξ, n) dβ_k(s)

η(ξ)dξ

2

L²_t,x

= 1

2π(δK)² Z

u

X

n

E

∞

X

k=1

Z ∞ 0

Z

R

ψ˜

L(iu, in, ξ) δK

e^−isu∂ξ(g\kχφ)_J(s, ξ, n)η(ξ)dξdβk(s)

2

du

= 1

2π(δK)² Z

u

X

n

E Z ∞

0

∞

X

k=1

Z

R

ψ˜

L(iu, in, ξ) δK

e^−isu∂ξ(g\kχφ)_J(s, ξ, n)η(ξ)dξ

2

dsdu

. 1 (δK)⁴

Z

u

X

n

E Z ∞

0

∞

X

k=1

Z

R

ψ˜⁰

L(iu, in, ξ) δK

Lξ(iu, in, ξ)

ϑ(ξ) (g\kχϑφ)_J(s, ξ, n)η(ξ)dξ

2

dsdu

+ 1

(δK)⁴ Z

u

X

n

E Z ∞

0

∞

X

k=1

Z

R

ψ˜

L(iu, in, ξ) δK

(g\kχφ)_J(s, ξ, n)η⁰(ξ)dξ

2

dsdu

Hence, by (2.3) it follows that

EkI3k²_L2 t,x

. 1 (δK)⁴E

Z ∞ 0

Z

u

X

n

Z

R

ψ˜⁰

L(iu, in, ξ) δK

Lξ(iu, in, ξ) ϑ(ξ)

2

1suppηdξ

× Z

R

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}

∞

X

k=1

(g\kχϑφ)_J(s, ξ, n)

2η²(ξ)dξduds

+ 1

(δK)⁴E Z ∞

0

Z

u

X

n

Z

R

ψ˜

L(iu, in, ξ) δK

2

1_suppηdξ

× Z

R

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}

∞

X

k=1

(g\kχϑφ)_J(s, ξ, n)

2|η⁰(ξ)|²dξduds

. 1 (δK)⁴

δK J^β

α

J^2β

×E Z ∞

0

Z

u

X

n

Z

R

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}

∞

X

k=1

(g\_kχϑφ)_J(s, ξ, n)

2η²(ξ)dξduds

+ 1

(δK)⁴ δK

J^{β α}

×E Z ∞

0

Z

u

X

n

Z

R

1{|u+b(ξ)·n|²+|n^∗A(ξ)n|²<(2δK)²}

∞

X

k=1

(g\kχϑφ)_J(s, ξ, n)

2|η⁰(ξ)|²dξduds

and due to (3.5) we obtain

EkI3k²_L2

t,x. 1 (δK)³

δK J^β

α

J^2βE Z ∞

0

X

n

Z

R

∞

X

k=1

(g\kχϑφ)_J(s, ξ, n)

2(η²+|η⁰|²) dξds

≤ 1 (δK)³

δK J^β

α

J^2β

∞

X

k=1

Ek(gkχϑφ)J(η+|η⁰|)k²_L2 t,x,ξ. (3.8)

Estimate ofI4. By Plancherel and Itˆo’s formula we have

EkI4k²_L2 t,x

= 1

2π(δK)²E Z

R

F_tx⁻¹

ψ˜

L(iu, in, ξ) δK

∞

X

k=1

Z ∞ 0

e^−isuFx((∂ξgk)χφ)_J(s, ξ, n) dβk(s)

η(ξ)dξ

2

L²_t,x

= 1

2π(δK)² Z

u

X

n

E

∞

X

k=1

Z ∞ 0

Z

R

ψ˜

L(iu, in, ξ) δK

e^−isuFx((∂ξgk)χφ)_J(s, ξ, n)η(ξ)dξdβk(s)

2

du

= 1

2π(δK)² Z

u

X

n

E Z ∞

0

∞

X

k=1

Z

R

ψ˜

L(iu, in, ξ) δK

e^−isuFx((∂ξgk)χφ)_J(s, ξ, n)η(ξ)dξ

2

dsdu